Finite Group Databasebeta

← #37536994

Subgroups — #37536994

Order 23168.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153781C211yesmaximal, characteristic, complemented
2C2×#19153774C211yesmaximal, characteristic
3#26041214C211yesmaximal, characteristic
4#26041213C212yesmaximal, complemented
5#26041203C212yesmaximal
6#26041199C211yesmaximal, characteristic, complemented
7#26041192C211yesmaximal, characteristic
8#26041130C211yesmaximal, characteristic, complemented
9#26041130C212yesmaximal, complemented
10#26041126C212yesmaximal
11#400×C181C211yesmaximal, characteristic, complemented
12C22×Dic362C2211yescharacteristic
13C22×C4×D181C2211yescharacteristic
14#19153781—24no—
15#19153774—24no—
16C2×C4⋊Dic181C2211yescharacteristic
17C2×#15668428C2212yes—
18C2×#15668428C2212yes—
19C2×#15668428C2212yes—
20C2×#15668428C2212yes—
21C2×#15668426C2212yes—
22C2×#15668426C2212yes—
23C2×#15668426C2211yescharacteristic
24C2×#15668426C2211yescharacteristic
25C2×C4×Dic181C2212yes—
26C2×C4×Dic181C2212yes—
27C2×C4×Dic181C2211yescharacteristic
28C2×C4×Dic181C2211yescharacteristic
29#19153723C2212yes—
30#19153723C2212yes—
31#19153723C2211yescharacteristic
32C2×C4⋊C4×C181C2211yescharacteristic
33C2×C4⋊C4×C181C2211yescharacteristic
34#95×C181C2212yes—
35#95×C181C2211yescharacteristic
36#95×C181C2212yes—
37C23×D181C2311yescharacteristic
38C2×Dic362—24no—
39C2×Dic362—22no—
40C22×Dic181C2311yescharacteristic
41C22×Dic181C2311yescharacteristic
42C22×Dic181C2312yes—
43C22×Dic181C2312yes—
44C22×Dic181C2311yescharacteristic
45C2×C4×D181—22no—
46C2×C4×D181—22no—
47C2×C4×D181Q814yescomplemented
48C4⋊Dic181—22no—
49#15668428—24no—
50#15668428—24no—
51#15668428—24no—
52#15668428—24no—
53#15668426—22no—
54#15668426—24no—
55#15668426—24no—
56#15668426—22no—
57C4×Dic181—22no—
58C4×Dic181—24no—
59C4×Dic181—24no—
60C4×Dic181D414yes—
61C22×C4×C181C2312yes—
62C22×C4×C181C2311yescharacteristic
63C22×C4×C181C2312yes—
64C22×C4×C181C2311yescharacteristic
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181—22no—
67C4⋊C4×C181—22no—
68C22×D181—21no—
69C22×D181—22no—
70C22×D181—22no—
71C22×D181—21no—
72C22×D181C2×Q812yes—
73Dic362—28no—
74C2×Dic181—22no—
75C2×Dic181—22no—
76C2×Dic181—22no—
77C2×Dic181—22no—
78C2×Dic181—22no—
79C2×Dic181—21no—
80C2×Dic181—22no—
81C2×Dic181C2×D412yes—
82C2×Dic181C2×D412yes—
83C2×Dic181C2×Q812yes—
84C2×Dic181#4114yes—
85C2×Dic181#4114yes—
86C2×Dic181#4112yes—
87C4×D181—28no—
88C23×C181C2411yescharacteristic
89C2×C4×C181—21no—
90C2×C4×C181—21no—
91C2×C4×C181—22no—
92C2×C4×C181—21no—
93C2×C4×C181—22no—
94C2×C4×C181—22no—
95C2×C4×C181—22no—
96C2×C4×C181—22no—
97C2×C4×C181—22no—
98C2×C4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181C2×D412yes—
101C2×C4×C181C2×Q812yes—
102C22×C181C22×D411yescharacteristic
103C22×C181C22×Q811yescharacteristic
104C22×C181C2×#4111yescharacteristic
105C22×C181C2×#4112yes—
106C22×C181C2×#4112yes—
107C2×D181—22no—
108C2×D181—24no—
109C2×D181—41no—
110C2×D181—42no—
111C2×D181—41no—
112C4×C181—22no—
113C4×C181—22no—
114C4×C181—41no—
115C4×C181—42no—
116C4×C181—42no—
117Dic181—24no—
118Dic181—24no—
119Dic181—22no—
120Dic181—41no—
121Dic181C22⋊Q814yes—
122C2×C181C2×C22⋊Q811yescharacteristic
123C2×C181#52511yescharacteristic
124C2×C181#52912yes—
125C2×C181#53211yescharacteristic
126C2×C181#53312yes—
127D181—42no—
128C181#239611yescharacteristic, complemented, Sylow
129#2396—1811nomaximal, Sylow
130C2×C4×Q8—1811no—
131C2×#117—1811no—
132#400D18111yescharacteristic, complemented
133#399—1811no—
134#398—1812no—
135#396—1812no—
136#395—1811no—
137#393—1811no—
138C4×C4⋊C4—1812no—
139C4×C4⋊C4—1811no—
140C4×C22⋊C4—1812no—
141C22×Q8—1811no—
142C23×C4—1811no—
143C4×Q8—3624no—
144#117—3624no—
145C2×C4⋊C4—1811no—
146C2×C4⋊C4—1812no—
147C2×C4⋊C4—1812no—
148C2×C4⋊C4—1812no—
149C2×C4⋊C4—1812no—
150C2×C4⋊C4C2×D18111yescharacteristic
151C2×C4⋊C4C2×D18111yescharacteristic
152C2×C22⋊C4—1811no—
153C2×C22⋊C4—1811no—
154C2×C22⋊C4—1812no—
155C2×C22⋊C4—1812no—
156C2×C42—1811no—
157C2×C42—1812no—
158C2×C42—1812no—
159C2×C42—1811no—
160#95—1812no—
161#95—1812no—
162#95—1811no—
163#95C2×D18112yes—
164#95C2×D18111yescharacteristic
165#95C2×D18112yes—
166C24—1811no—
167C2×Q8—3624no—
168C2×Q8—3622no—
169C22×C4—3622no—
170C22×C4—3622no—
171C22×C4—1811no—
172C22×C4—1812no—
173C22×C4—1812no—
174C22×C4—1811no—
175C22×C4—1814no—
176C22×C4—1811no—
177C22×C4C22×D18112yes—
178C22×C4C22×D18111yescharacteristic
179C22×C4C22×D18112yes—
180C22×C4C22×D18111yescharacteristic
181C22×C4C22×D18111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—3622no—
185C4⋊C4—3624no—
186C4⋊C4—3624no—
187C4⋊C4—3624no—
188C4⋊C4—3624no—
189C22⋊C4—3622no—
190C22⋊C4—3624no—
191C22⋊C4—3624no—
192C22⋊C4—3622no—
193C42—3622no—
194C42—3624no—
195C42—3624no—
196C42—1814no—
197C23—3621no—
198C23—3622no—
199C23—3622no—
200C23—3621no—
201C23—1812no—
202C23C23×D18111yescharacteristic
203Q8—3628no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—3628no—
216C2×C4—3622no—
217C2×C4—3622no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4—3622no—
221C2×C4—3621no—
222C2×C4—3622no—
223C2×C4—1812no—
224C2×C4—1814no—
225C2×C4—1814no—
226C2×C4—1812no—
227C2×C4—1812no—
228C2×C4—1812no—
229C2×C4D4×D18112yes—
230C2×C4Q8×D18112yes—
231C22—3622no—
232C22—3624no—
233C22—7241no—
234C22—7242no—
235C22—7241no—
236C22C2×D4×D18111yescharacteristic
237C22C2×Q8×D18111yescharacteristic
238C22#41×D18111yescharacteristic
239C22#41×D18112yes—
240C22#41×D18112yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—3624no—
247C4—3624no—
248C4—3622no—
249C4—1814no—
250C4—7241no—
251C2—7242no—
252C2C22⋊Q8×D18111yescharacteristic
253C2#2604178711yescharacteristic
254C2#2604181212yes—
255C2#2604182811yescharacteristic
256C2#2604183512yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#400×C181C2
prime radicalC23×D181C23
hypercenter#400D181
hypocenterC181#2396
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153781C2semidirect
4. #26041213C2semidirect
6. #26041199C2semidirect
8. #26041130C2semidirect
9. #26041130C2semidirect
11. #400×C181C2semidirect
47. C2×C4×D181Q8semidirect
128. C181#2396semidirect
132. #400D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191537812
C2×#191537742
#260412142
#260412132
#260412032
#260411992
#260411922
#260411302
#260411262
#400×C1812
#2396181